where \(X\sim p\)  and the kernels \(K\)  and \(k\)  above are defined in Definition 1 (see [1]}).
We say that a weakly differentiable function \(h\)  is increasing if \(h^{\prime }\ge 0\)  a.e. The same convention holds for a monotone function.
We recall that \(L^2(p)\)  and \(L^2(pw)\)  are
separable Hilbert spaces (see [2]}) and define the Sobolev space
\(H^1(p,w)=\left\lbrace h\in L^2(p): h\text{ is weakly differentiable and}h^{\prime }\in L^2(pw) \right\rbrace .\) 
